Question Statement

Find $\frac{dy}{dx}$ given that:

$y=e^{4x}(1+\ln x)$

---

Background and Explanation

This problem involves differentiating a product of two functions: an exponential function $e^{4x}$ and a logarithmic expression $(1+\ln x)$. You will need to apply the product rule along with the chain rule for the exponential term and the standard derivative for the natural logarithm.

---

Solution

We are given $y=e^{4x}(1+\ln x)$. Since this is a product of two functions, we apply the product rule: $\frac{d}{dx}[u\cdot v]=u^{\prime }v+u{v}^{\prime }$.

Step 1: Set up the product rule with $u=e^{4x}$ and $v=(1+\ln x)$.

$\frac{dy}{dx}=\frac{d}{dx}\left(e^{4x}\right)\cdot (1+\ln x)+e^{4x}\cdot \frac{d}{dx}(1+\ln x)$

Step 2: Differentiate each part.

• For the first term, use the chain rule on $e^{4x}$: $\frac{d}{dx}(e^{4x})=e^{4x}\cdot \frac{d}{dx}(4x)=4e^{4x}$
• For the second term: $\frac{d}{dx}(1+\ln x)=0+\frac{1}{x}=\frac{1}{x}$

Substituting these back:

$\frac{dy}{dx}=e^{4x}\cdot \frac{d}{dx}(4x)\cdot (1+\ln x)+e^{4x}\cdot \left(0+\frac{1}{x}\right)$

Step 3: Simplify the derivatives.

$\frac{dy}{dx}=e^{4x}\cdot 4\cdot (1+\ln x)+e^{4x}\cdot \frac{1}{x}$

Step 4: Factor out the common term $e^{4x}$ to present the final simplified answer.

$\frac{dy}{dx}=e^{4x}\left(4(1+\ln x)+\frac{1}{x}\right)$

---

Key Formulas or Methods Used

• Product Rule: $\frac{d}{dx}[u(x)v(x)]=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$
• Chain Rule: $\frac{d}{dx}[e^{u(x)}]=e^{u(x)}\cdot u^{\prime }(x)$
• Derivative of Natural Logarithm: $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• Constant Rule: $\frac{d}{dx}(c)=0$

---

Summary of Steps

1. Identify the structure: Recognize $y$ as a product of $e^{4x}$ and $(1+\ln x)$
2. Apply product rule: $\frac{dy}{dx}=\frac{d}{dx}(e^{4x})\cdot (1+\ln x)+e^{4x}\cdot \frac{d}{dx}(1+\ln x)$
3. Differentiate $e^{4x}$: Use chain rule to get $4e^{4x}$
4. Differentiate $(1+\ln x)$: Get $0+\frac{1}{x}=\frac{1}{x}$
5. Substitute and expand: $4e^{4x}(1+\ln x)+\frac{e^{4x}}{x}$
6. Factor: Extract $e^{4x}$ to obtain final form: $e^{4x}\left(4(1+\ln x)+\frac{1}{x}\right)$